An Obata-type characterization of doubly-warped product Kahler manifolds

We give a characterization {sl `a la Obata} for certain families of Kahler manifolds. These results are in the same line as other extensions of the well-known Obatas rigidity theorem from cite{Obata62}, like for instance the generalizations in cite{RanjSant97} and cite{Santhanam07}. Moreover, we give a complete description of the so-called Kahler doubly-warped product structures whose underlying metric is Einstein.